Conjecture on the number of linear relations in length two

Let $k\geq 4$. Consider the linear relations among the generators of the weight-$k$, length-two component $\operatorname{gr}^{\operatorname{W},\operatorname{L}}_{k,2}\operatorname{q\mathcal{M}\mathcal{Z}}$. The relation-count conjecture. For every weight $k\geq 4$, the number of such linear relations equals

$$\left\lfloor\frac{k}{2}\right\rfloor-1.$$

The authors verified the claim for $k\leq 20$, where this method produced all observed length-two relations; the assertion is proposed on the basis of that computation.