Syzygy conjecture for general projections of canonical curves

Let $C\hookrightarrow\mathbb{P}^{g-1}$ be a generic canonical curve of genus $g\geq 9$, and let $p\in\mathbb{P}^{g-1}$ be a generic point. Write $C'$ for the projection of $C$ away from $p$ to a hyperplane, and let $\widetilde{b}_{t,1}$ denote the graded Betti numbers of its syzygy resolution. Then the canonical projection conjecture. The curve $C'$ has the following properties: (1) it satisfies $\widetilde{N}_{\operatorname{Cliff}(C)-3}$; (2) the length of the linear strands of its syzygy resolution is $g-\operatorname{Cliff}(C)-3$, meaning that $\widetilde{b}_{t,1}=0$ if and only if $1\leq t\leq g-\operatorname{Cliff}(C)-3$; and (3) if $g=2k$, then $\widetilde{b}_{k-2,1}=\widetilde{b}_{k-3,2}=1$, with the extra syzygy having maximal rank. These predictions refine the expected syzygies of general projections and remain conjectural in the stated generality.