The identification conjecture for $B^1_{g,\overline{d}}$ and $A^1_{g,\overline{d}}$

For $g\ge 0$, $n\ge 1$, and $\overline{d}=(d_1,\ldots,d_n)\in\mathbb Z_{\ge 0}^n$ with $\sum d_i\ge 2g$, let $B^1_{g,\overline{d}}$ and $A^1_{g,\overline{d}}$ be the tautological classes defined in $R^{\sum d_i}({\overline{\mathcal{M}}}_{g,n+1})$ from the balanced-tree construction and the coefficients of the double-ramification expression, respectively.

Identification conjecture. For any such $g$, $n$, and $\overline{d}$,

$$B^1_{g,\overline{d}}=A^1_{g,\overline{d}}\quad\text{in }R^{\sum d_i}({\overline{\mathcal{M}}}_{g,n+1}).$$

The $n=1$ case appeared previously, and the supplied text notes a proof in the Gorenstein quotient; the full equality in the tautological ring is not stated as resolved.