Trivariate shuffle conjecture for selected hook shapes

Let $\rho=(a\,|\,b)$ be a hook, let $\Gamma_a$ be the Dyck path defined in the source, and let $\mathbb{L}_\beta(t;\bm{z})$ be the LLT polynomial associated with a Dyck path $\beta$. Order Dyck paths by the Tamari poset, and let $d(\alpha,\beta)$ be the length of the longest strict chain from $\alpha$ to $\beta$. Trivariate shuffle conjecture. If $a$ is $0$, $1$, or $n-1$, then

$$\mathcal{S}_\rho(q,t,1;\bm{z})=\sum_{\Gamma_a\leq\alpha\leq\beta}q^{d(\alpha,\beta)}\mathbb{L}_\beta(t;\bm{z}),$$

where $\alpha$ lies below $\beta$ in the Tamari poset. The claim extends the trivariate shuffle conjecture cited by the source for $\rho=(0\,|\,n-1)$; the source gives no general resolution.