A 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. Zero-part formula. After inserting $\nu_1=0$,

$$\int_{\operatorname{logDR}_0(d,-\boldsymbol{\nu})} \psi_1^{e}\cdot \operatorname{branch}_{m-1-e}\big|_{\nu_1=0} =(m-1-e)\frac{(m-2)!}{(e+1)!} \cdot \Big( \sum_{i=1}^{e+1} \prod_{j=0}^{m-e-3}(d-(i+j)\frac{k}{2})\Big).$$

The formula is introduced as a conjectural expression obtained by inserting a contracted marked point into the recursion. The text notes that it currently has no geometric meaning in this form, while suggesting that it should acquire one.