Konvalinka–Tewari conjecture on the parking space extension

Let $n$ be a positive integer. Let $\tau_{n,1}$ denote the $S_n$-representation constructed from the authors' extension of the parking space at parameter $c=1$, and let $\rho_{n-1}$ denote the classical parking function representation of $S_{n-1}$. Let $\sigma_n$ denote the ungraded extension of the parking function representation constructed by Berget and Rhoades. For a representation $V$ of $S_n$, write $\mathrm{Frob}(V)$ for its Frobenius characteristic, and call $V$ $h$-positive if $\mathrm{Frob}(V)$ expands positively in the basis of homogeneous symmetric functions.

Konvalinka–Tewari conjecture. The representation $\tau_{n,1}$ is isomorphic to $\sigma_n$. Furthermore, $\mathrm{Frob}(\tau_{n,1})$ is $h$-positive.

This conjecture proposes that the authors' natural extension at $c=1$ agrees with the ungraded extension introduced by Berget and Rhoades, while also having positive homogeneous-symmetric-function expansion. The source presents both assertions as conjectural based on empirical evidence; no resolution is given.