Loehr–Warrington rational parking-function Frobenius conjectures

Let $\mathsf{PF}_{a,b}(q,t)$ be the symmetric function defined from rational parking functions, and let ${\sf Sch}_{a,b;k}(q,t)$ denote the coefficient of $s_{(k+1,1^{a-k-1})}$ in it. Rational parking-function Frobenius conjectures. The following properties are conjectured:

1. $\mathsf{PF}_{a,b}(q,t)=\mathsf{PF}_{a,b}(t,q)$.
2. $\langle q^{(a-1)(b-1)/2}\mathsf{PF}_{a,b}(q,1/q),h_{(1^a)}\rangle=[b]_q^{a-1}$.
3. For the hook coefficient,

$$q^{(2ak-k-k^2+ba-a^2-b+1)/2}{\sf Sch}_{a,b;k}(q,1/q)=\frac{1}{[b]_q}\genfrac{[}{]}{0pt}{}{a-1}{k}_q\genfrac{[}{]}{0pt}{}{b+k}{a}_q.$$

These conjectures seek symmetry and specializations for the rational parking-function Frobenius series; the source presents them as conjectural and gives no resolution.