The esv conjecture for B-cycle and modular graph functions

Let $\Gamma$ be a graph with $l$ edges. Define the Laurent polynomial

$$\mathbf{b}[\Gamma]:=\sum_{k=-l}^l b_k^{(0)}(\Gamma)\,T^k$$

and let $\mathbf{d}[\Gamma]$ denote the corresponding Laurent polynomial for the modular graph function. Define $\operatorname{esv}:\mathcal{Z}[T]\rightarrow\mathcal{Z}^{\rm sv}[y]$ by

$$\operatorname{esv}(\zeta(\mathbf{k}))=\zeta_{\rm sv}(\mathbf{k}),\qquad \operatorname{esv}(T)=-2y.$$

Brödel–Schlotterer–Zerbini's esv conjecture. For all graphs $\Gamma$,

$$\operatorname{esv}(\mathbf{b}[\Gamma])=\mathbf{d}[\Gamma].$$

The identity is supported by explicit computations and relates the constant Fourier modes of open-string B-cycle graph functions to modular graph functions; its validity for all graphs remains open.