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Let's suppose the underlying S follows a BS dynamic with the drift being the short rate that follows a short dynamic model.

the "local volatility" of the equity should be the implied volatility from the market (of the forward) corrected from the volatility of the short rate.

Now let's suppose that for [0,T1] and [0,T2] I have deduced the "local volatility" like described above.

Can I use the fact that the variance is additive to the local volatility to deduce the local volatility in [T1,T2].

Or is it only the variance of the forward that is additive?

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