The cohomology and ring structure conjecture for generic odd linear-tree varieties

Let $n$ be odd, and let $X_n=X_{\mathbb{A}_n}^{\mathtt{generic}}$ be the generic fiber associated with the linear tree $\mathbb{A}_n$. Write $\operatorname{WP}$ for the distinguished cohomology class in degree $2$. Cohomology conjecture. The Hodge structure on the cohomology of $X_n$ is given by

$$\mathsf{H}^k(X_n)=\mathbb{Q}(-k)$$

for even $k$ with $0\leq k\leq n-1$, and

$$\mathsf{H}^{n}(X_n)=\bigoplus_{i=(n+1)/2}^{n}\mathbb{Q}(-i).$$

The cohomology ring has a basis consisting of the powers $\operatorname{WP}^i$ for $0\leq i\leq (n-1)/2$ together with a basis of $\mathsf{H}^{n}(X_n)$, and it is generated by $\operatorname{WP}$ in degree $2$ and the elements of $\mathsf{H}^{n}(X_n)$ in degree $n$. This conjectural description predicts the Hodge structure and ring generators for the generic fibers associated with odd linear trees; the supplied text does not state a resolution, so the conjecture is recorded as open.