discussed in [@AF] and we briefly review it in Section 2. In particular, it predicts the following:

*Maximal Rank Conjecture $(MRC)_{g, d}^r$: * We fix integers $g, r, d\geq 1$ such that $$0\leq \rho(g, r, d)< 2d+2-g-{r+2\choose 2}.$$ For a general curve $[C]\in \cM_g$, the map $\nu_2(l):\mbox{Sym}^2(V)\rightarrow H^0(C, L^{\otimes 2})$ is injective for every linear series $l=(L, V)\in G^r_d(C)$.

Returning to Theorem \[countermercat\], $(MRC)_{g, 2a+3}^4$ predicts that the syzygy locus $\mathfrak{Syz}_{g, 2a+3}^4$ is a proper subvariety of $\cM_g$, and then it must be a divisor.

\[mrc4sectiuni\] Fix an integer $a\geq 5$ and a general curve $[C]\in \cM_{2a+1}$. Then $$K_{1, 1}(C, L)=0\ \mbox{ for every }\ L\in W^4_{2a+3}(C),$$ and the failure locus $\mathfrak{Syz}_{g, 2a+3}^4$ is a divisor in $\cM_{2a+1}$. Consequently, Mercat’s Conjecture (\[mercat2\]) holds for all curves in the complement of $\mathfrak{Syz}_{g, 2a+3}^4$.

Using Mukai’s work [@M1], we can confirm this expectation in one interesting case, namely that of curves of genus $11$, and answer Question 5.5 in [@LMN]:

\[gen11\] The locus $\mathfrak{Syz}_{11, 13}^4:=\{[C]\in \cM_{11}: \exists L\in W^4_{13}(C) \ \mathrm{with}\ K_{1, 1}(C, L)\neq 0\}$  is an effective divisor in $\cM_{11}$. In particular, $\mathcal{BN}_C(13, 4)=\emptyset$ for a general curve $[C]\in \cM_{11}$.

The above mentioned relation to syzygies, enables us to prove conjecture 