An alternative zero-part formula for logarithmic double ramification intersections

Let $d$ be the degree, let $\boldsymbol{\nu}$ be the ramification data, and let $\operatorname{logDR}_0(d,-\boldsymbol{\nu})$ denote the logarithmic double ramification cycle. Let $\psi_1$ be the cotangent-line class at the first marking and let $\operatorname{branch}_{m-1-e}$ be the corresponding branch class. Alternative zero-part formula. After inserting $\nu_1=0$,

$$\int_{\operatorname{logDR}_0(d,-\boldsymbol{\nu})} \psi_1^{e}\cdot \operatorname{branch}_{m-1-e}\big|_{\nu_1=0} = \frac{(m-2)!}{(e+1)!}\cdot \frac{\prod_{j=e+1}^{m-1}(d-j\frac{k}{2}) -\prod_{j=0}^{m-e-2}(d-j\frac{k}{2}) }{(-k/2)}.$$

This is presented as an alternative form of the preceding conjectural zero-part formula, closer to its $k=0$ specialization. It is therefore a restatement rather than a separate mathematical conjecture.