Using converted lognormal volatilities for negative rates in a lognormal Libor Market Model (LMM)

There exist formulas to convert between normal and lognormal interest rate volatilities. In the most simple form the approximation for ATM volatilities would be $$\sigma_{LogNorm}=\frac{\sigma_{Norm}}{\text{|forward rate|}}$$. Such conversion makes it possible to calculate $$\sigma_{LogNorm}$$ also for negative rates, for which the normal volatility exists but the lognormal Black volatility doesn't.

The question is: Could these converted lognormal volatilities be used in a lognormal Libor Market Model (LMM) for modelling negative interest rates? Or is there something fundamentally wrong with such approach?

I believe the correct/standard approach would be to use a shifted lognormal LMM, but I currently only have access to a lognormal BGM model.

It seems that the LMM dynamics for which your $$\sigma_{LogNorm}$$ is valid would be $$dF_t = |F_t|\sigma_{LogNorm}dW_t$$ rather than the usual description. So it would not be a consistent volatility to use in your model when rates are negative.