options - Delta Hedging with fixed Implied Volatility to get rid of vega?

I'm wondering if i should use a floating IV or a fixed IV to delta hedge my options every day.

I've read this post but would like different information : Delta Hedging with fixed Implied Volatility or floating Implied Volatility?

I have a view on future realized volatility for an option $\sigma_{e}$ (e for expected).
When i look at an option, if its implied volatility $\sigma_{i0}$ is lower than $\sigma_{e}$ then i want to buy the option and delta hedge at a certain frequency (daily for example).

I've always delta hedged using a floating IV which is changing daily but i realized this may not best thing to do.

If we decompose the variation of a PNL between 2 hedging periods we have :

$$ dPNL = \vartheta * d\sigma + \theta*dt + 0.5 * dS^2*\Gamma $$

I've considered as well known that the daily PNL of a delta hedged between option is given by :
$$ dPNL = 0.5 *(\sigma_{i}^2-RV^2)\Gamma * S * dt $$ with $RV$ = realized volatility = $ ds/S $

My concern with this is that i have the feeling that we are only looking at $\Gamma$ and $\theta$ influences in the case we are delta hedging using floating IV every day.

Looking at this formula, PNL at maturity should not be IV path dependant, however running simulations i get to very different PNL at maturity using a floating IV. So is this formula only true when using a fixed IV to hedge daily ?

My intuition is the following :
If we want to get rid of the vega effect then we need to replicate the same option with a constant IV so that the vega does not have any effect on PNL.

But does this mean that we can use any IV to hedge the option ?
This makes no sense to me ?
I would like to have a mathematical proof of this coming from the Greek PNL decomposition above if possible ?

What IV should i use in my situation ?