Shahriari and Yu's conjecture for forbidden copies of $Y_k$ and $Y'_k$

Let $q$ be a prime power, and let $\mathrm{La}_q(n,\{Y_k,Y'_k\})$ denote the maximum size of a family of subspaces of an $n$-dimensional vector space over $\mathbb{F}_q$ containing neither $Y_k$ nor $Y'_k$ as a subposet. Let $\Sigma_q(n,k)$ denote the corresponding sum of the $k$ largest Gaussian binomial coefficients. Shahriari and Yu's conjecture. For $k\geq 1$ and $n\geq k+1$,

$$\mathrm{La}_q(n,\{Y_k,Y'_k\})=\Sigma_q(n,k).$$

The conjecture is a vector-space analogue of a forbidden-poset theorem in the Boolean lattice; the cited paper proves this result using the $q$-analogue of the Lubell–Yamamoto–Meshalkin inequality.