sections of $K3$ surfaces lying in certain Noether-Lefschetz loci. For the curves appearing in Theorem \[countermercat\], we observe that $\beta_g(2, d, 4)=-3$. The possibility that Mercat’s Conjecture might fail for some curves of genus $11$ was already entertained in [@GMN] Remark 3.5 and [@LMN] Question 5.5. In fact, it was that particular suggestion in loc. cit. that drew our attention to this problem.

The proof of Theorem \[countermercat\] uses again the observation that for a curve $C$ of genus $2a+1$ and gonality $a+2$, if $L\in W^4_{2a+3}(C)$ is a linear series such that the multiplication map $\nu_2(L):\mbox{Sym}^2 H^0(C, L)\rightarrow  H^0(C, L^{\otimes 2})$ is not injective, then $\mathcal{BN}_C(2a+3, 4)\cap \mathcal{SU}_C^s(2, L)\neq  \emptyset$. More precisely, the locus of curves $[C]\in \cM_{2a+1}$ with $\mathcal{BN}_C(2a+3, 4)\neq \emptyset$ is set-theoretically equal to the Koszul locus $$\mathfrak{Syz}_{g, 2a+3}^4:=\{[C]\in \cM_{2a+1}: \exists L\in W^4_{2a+3}(C)\ \mbox{ such that } K_{1, 1}(C, L)\neq \emptyset\}.$$ This is a virtual divisor in $\cM_{2a+1}$, which *is not contained* in the Hurwitz divisor [@HM] $$\cM_{2a+1, a+1}^1:=\{[C]\in \cM_{2a+1}: W^1_{a+1}(C)\neq \emptyset\}$$ of curves with a $\mathfrak g^1_{a+1}$. Curves $[C]\in \mathfrak{Syz}_{g, 2a+3}^4-\cM_{g, a+1}^1$ provide counterexamples to (\[mercat2\]).

Even though there curves of maximal Clifford index not verifying (\[mercat2\]), the question whether Mercat’s inequalities (\[mercat1\]) and (\[mercat2\]) are true 