Lin and Kim's refinement conjecture for Schröder numbers

Let $S_{n,k}$ be the triangle defined by

$$S_{n,k}=S_{n,k-1}+2S_{n-1,k}-S_{n-1,k-1},$$

for $1\leq k\leq n-2$, with $S_{n,n}=S_{n,n-1}=S_{n,n-2}$ for $n\geq 3$ and $S_{1,1}=S_{2,1}=S_{2,2}=1$. For a pair of patterns $(\nu,\mu)$, let $\mathcal{S}_n(\nu,\mu)$ denote the permutations in $\mathcal{S}_n$ avoiding both patterns. Lin and Kim's conjecture. Let $(\nu,\mu)$ be a pair of patterns of length four. Then

$$S_{n,k}=\left|\{\sigma_1\cdots\sigma_n\in\mathcal{S}_n(\nu,\mu)\mid \sigma_n=k\}\right|$$

for all $1\leq k\leq n$ if and only if $(\nu,\mu)$ is one of the following nine pairs:

$$(4321,3421),\ (3241,2341),\ (2431,2341),\ (4231,3241),\ (4231,2431),$$ $$(4231,3421),\ (2431,3241),\ (3421,2431),\ (3421,3241).$$

Lin and Kim's conjecture proposes a precise refinement of the Schröder-number enumeration, identifying exactly the two-pattern avoidance classes whose last-entry distribution is given by the triangle $S_{n,k}$. The source presents it as a conjecture, and no resolution is supplied here.