RFR boostrapping using RFR OIS: Is convexity adjustment technically necessary?

For single-curve RFR bootstrapping, such as a SOFR-based discounting curve bootstrapped strictly using SOFR fixed-float OIS, I am trying to understand if convexity adjustments are technically necessary in the bootstrapping process to calculate the present values of the OIS coupons due to the payment lag (delay after the fixing period end date) that is exhibited in such swaps, e.g. a SOFR fixed-float swaps pays coupons two good business days from the end of coupon fixing periods.

I have searched through quant.stackexchange and academic and practitioner literature for insights into this question and have not been entirely satisfied with the results, so thus the question here.

It seems that most of the discussions on quant.stackexchange over the past few years related to the need for convexity adjustment in swaps seem to primarily bei in the context of Libor in arrears (e.g. here and here). This question on OIS curve convexity adjustment, is somewhat is asking the question above, but I feel that the answers do not entirely answer the question.

In the literature, the concept of convexity adjusting coupons of OIS is addressed in Henrard (2004) and also in chapter 6 of Henrard 2014, where Henrard demonstrates via a HJM framework (in particular, a single-factor Hull-White model) that the impact of a two day payment lag over a 1-year period is less than 0.003 bps in rate per \$1 bn notional. From this analysis, Henrard concludes that the impact is small enough to be omitted. That being said, I am unclear from Henrard's discussions on this topic if such convexity adjustments are technically necessary for boostrapping. Also, Ametrano and Bianchetti (2013) omit payment lag entirely in their very detailed exposition of the mechanics of OIS leg payments in the context of multi-curve bootstrapping.

I have not been able to find any detailed literature on the LCH or CME SOFR curves; however, I have been able to replicate Bloomberg's OIS bootstrapping methodology (e.g. used in curves 42 USD OIS and 490 SOFR) under Bloomberg's 3rd ICVS interpolation method of step-function forward with continuously-compounded forward rates. It seems pretty clear from my calculations that the tenor nodes that Bloomberg uses in their OIS bootstrapping routine are the maturity (accrual end) dates of OIS with tenors equal to or less than 12-months, and and payment dates (accrual end dates + 2) tenors 18-months and greater. For the swaps with more than one payment it seems like payments dates are being considered for each of a given swap's coupons. Overall, Bloomberg's calculation of OIS floating rate coupons for tenors 18M and greater is consistent with section C.4 of Ametrano and Bianchetti (2013) using the authors' Eq. (148) but fixed and float payments are discounted at the payment date rather than the end date of the fixing. For example, this can be observed in SWPM's cashflow tables for a particular OIS while referencing a particular valuation date's OIS curve. Concretely, referencing Eq. 143 in Ametrano and Bianchetti (2013), the price of a OIS coupon payment, or $${\rm\bf OISlet}$$, with accrual period $$[T_{i-1},T_i]$$ at time $$t is

$$\begin{eqnarray} {\rm\bf OISlet}_{\rm float}(t;T_{i-1},T_i,R_{on}) &:=& P_c(t;T_i)\mathbb{E}^{Q_f^{T_i}}_i\left[{\rm\bf OISlet}_{\rm float}(t;T_{i-1},T_i,R_{on})\right]. \end{eqnarray}$$

It seems that Bloomberg is doing the following to account for the payment lag:

$$\begin{eqnarray} {\rm\bf OISlet}_{\rm float}(t;T_{i-1},T_i,R_{on}) &:=& P_c\left(t;T_i^{\color{red}{Payment}}\right)\mathbb{E}^{Q_f^{T_i}}_i\left[{\rm\bf OISlet}_{\rm float}(t;T_{i-1},T_i,R_{on})\right]. \end{eqnarray}$$

I am unclear if this is incorrect or not, but it seems from the literature that technically the the expectation would need to be changed from $$T_i$$ to $$T_i^{\rm Payment}$$ and this measure change would result in a convexity adjustment such as that discussed in Henrard. However, since rates are ultimately deterministic here, I am unclear if such a adjustment is necessary or not.

Anyway, thank you for your time, and any advice or help here would be appreciated.