Restricted Marstrand projection conjecture for analytic families

Let $P_t:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be an analytic family of linear maps, meaning that its coordinate functions are analytic functions of $t$. Assume that for every subspace $\pi$ of $\mathbb{R}^n$, one has

$$\dim P_t(\pi)=\min(\dim\pi,m)$$

for all but finitely many $t\in[0,1]$. Restricted Marstrand's conjecture. For every Borel set $A\subset\mathbb{R}^n$,

$$\dim_H P_t(A)=\min(\dim_H A,m)$$

for almost every $t\in[0,1]$. This conjecture asserts that failure of the restricted Marstrand projection theorem should already be detected by a subspace. It generalizes the expected almost-everywhere Hausdorff-dimension formula from classical projection theory to analytic families of linear maps.