Symmetric-algebra description of the local line Quot vanishing-cycle cohomology

Let $Q_L^n$ be the Quot schemes associated with the local line $L\subset\mathbb A^3$, and let $\phi^{\operatorname{mon}}_{\operatorname{Tr}(W_r)}\underline{\mathbb Q}_{\mathcal N^{\circ\circ}_n}$ denote the relevant monodromic vanishing-cycle complex. Let $\mathbb T$ be the Tate object, and let $V$ be a one-dimensional pure weight-zero Hodge structure placed in degree $1$. Symmetric-algebra conjecture. There is an isomorphism of $\mathbb Z$-graded mixed Hodge structures

$$\bigoplus_{n\geq 0}\operatorname{H}_c(Q_L^n,\phi^{\operatorname{mon}}_{\operatorname{Tr}(W_r)}\underline{\mathbb Q}_{\mathcal N^{\circ\circ}_n})\otimes\mathbb T^{-2n-n^2} \cong \operatorname{Sym}\left(\bigoplus_{n\geq 1}\left(\bigoplus_{i=-2}^{-n-1}\mathbb T^i\right)\right)\otimes\operatorname{Sym}(V),$$

with $n$ tracking the degree on both sides; the grading on the right comes from the graded object inside the symmetric algebra. The source asserts that the equivalent purity and isomorphism statements are true, so this is a solved theorem rather than an open conjecture.