Generalized restricted Marstrand conjecture for analytic linear families

Let $P_t:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be an analytic family of linear maps. Define $f:[0,n]\rightarrow\mathbb{R}$ as follows: for an integer $x\in[0,n]$, let $f(x)$ be the largest integer such that every subspace $A\subset\mathbb{R}^n$ with $\dim A=x$ satisfies

$$\dim P_t(A)\geq f(x)$$

for all but finitely many $t\in[0,1]$; for noninteger $x$, define $f$ linearly on $[\lfloor x\rfloor,\lfloor x\rfloor+1]$. Generalized restricted Marstrand conjecture. For every Borel set $A\subset\mathbb{R}^n$,

$$\dim_H P_t(A)\geq f(\dim_H A)$$

for almost every $t\in[0,1]$. This is a more general lower-bound formulation of the restricted Marstrand problem, allowing the expected projection dimension to be determined by the eventual behavior of the family on subspaces.