Scrollar invariants from standard Young tableaux on Hirzebruch surfaces

Let $F_e$ be the Hirzebruch surface with ruling $F_e\to\mathbf{P}^1$, let $e\geq 0$, and let $C$ be a smooth curve on $F_e$ with

$$C\sim dE+(c+de)F,$$

where $d\geq 2$ and $c\geq 0$. Let $\varphi:C\to\mathbf{P}^1$ be the map induced by the ruling, and assume that it is simply branched. For a partition $\lambda\vdash d$, let $e(T)$ be the integer attached to a standard Young tableau $T$ of shape $\lambda$ by summing the scrollar invariants $e_i$ over the indices $i$ for which $i+1$ appears to the left of $i$ in the reading word of $T$.

Scrollar-invariant tableau conjecture. The multiset of scrollar invariants of the partition $\lambda$ with respect to $\varphi$ is

$$\{e(T)\mid T\text{ is a Young tableau of shape }\lambda\}.$$

The assertion proposes a representation-theoretic description of scrollar invariants using standard Young tableaux. The supplied passage gives the geometric hypotheses and the proposed formula, but no resolution or partial result for this statement.