Critical-parameter limit for the measure of t-intersecting subspaces

Let $t$ be a fixed positive integer. Let $\Omega_n$ be the set of all subspaces of an $n$-dimensional vector space over the field with $q$ elements, let $A_n^{(t)}$ be the family of subspaces containing a fixed $t$-dimensional subspace, and define

$$f(n,t,\sigma)=\max\\{\mu_{\sigma}(U)^{1/n}: U\subset\Omega_n\text{ is }t\text{-intersecting}\\}.$$

Write $\sigma_{\theta,n}=q^{-(1-\theta)n}$ and $\mu_{\theta,n}=\mu_{\sigma_{\theta,n}}$. The critical-parameter limit conjecture.

$$\lim_{n\to\infty}f(n,t,\sigma_{\frac{1}{2},n})=q^{-\frac{1}{2}t}.$$

The theorem immediately preceding this statement establishes the corresponding limits away from the critical value $\theta=\frac12$; this conjecture concerns the remaining boundary case and is presented as the expected critical asymptotic.