The representation-number doubling conjecture for maximal lattices

Let $\mathcal V$ be a totally positive definite quadratic space over a totally real number field $k$ of dimension $g$. Fix a maximal integral lattice $L$ of $\mathcal V$, and let $T\in\frac{1}{2}\mathcal E_g(\mathfrak o)$ be totally positive definite. Write $R(L,T)$ for the representation number of $T$ by $L$. If $g$ is even and $\chi^{\mathcal V}=1$, then there is a totally positive definite matrix $T'\in\frac{1}{2}\mathcal E_{g-1}(\mathfrak o)$ such that

$$R(L,T)=2R(L,T').$$

Representation-number doubling conjecture. Under these hypotheses, such a matrix $T'$ exists and the representation number satisfies the displayed doubling identity. The claim concerns a relation between representation numbers of a maximal lattice and totally positive definite matrices of consecutive sizes; the supplied text does not indicate whether it has been proved or remains open.