Parametric Khintchine-like behaviour conjecture for iterated function systems

Let $U$ be an open subset of $\mathbb{R}^k$, and for each $u\in U$ let $\Phi_u=\{\phi_{i,u}(x)=A_i(u)x+t_i(u)\}_{i=1}^l$ be an iterated function system with attractor $X_u$. Let $\mathfrak m$ be a slowly decaying $\sigma$-invariant ergodic probability measure, with entropy $\mathfrak h(\mathfrak m)$ and Lyapunov exponents $\lambda_1(\mathfrak m,u),\ldots,\lambda_d(\mathfrak m,u)$. Write $H^*$ for the class of functions used in the source and $U_{\Phi_u}(z,\mathfrak m,h)$ for the corresponding limsup set. Parametric Khintchine-like behaviour conjecture. If

$$\mathfrak h(\mathfrak m)>-\bigl(\lambda_1(\mathfrak m,u)+\cdots+\lambda_d(\mathfrak m,u)\bigr)$$

for Lebesgue almost every $u\in U$, then, under some weak assumptions on the maps $A_i$ and $t_i$, both stated conclusions hold: for almost every $u\in U$, every $z\in X_u$ and $h\in H^*$ give $U_{\Phi_u}(z,\mathfrak m,h)$ full Lebesgue measure in $X_u$; and for almost every $u\in U$ and every $z\in X_u$, some $h:\mathbb N\to[0,\infty)$ with $\sum_{n=1}^{\infty}h(n)=\infty$ gives a set $U_{\Phi_u}(z,\mathfrak m,h)$ of zero Lebesgue measure. This is proposed as a generalisation from the parameterised families accessible to the paper's methods; the required weak assumptions are not specified.