Conjecture on parity-dependent convergence of periodic spline projections

Let ${\boldsymbol \tau}$ be a uniform knot sequence such that the dimension of $\mathcal{S}_{p,{\boldsymbol \tau},\operatorname{per}}$ is $2m$. For any $q\geq 0$, let $Q_p^q$ be the projection onto $\mathcal{S}_{p,{\boldsymbol \tau},\operatorname{per}}$ defined by the periodic projection construction. Parity convergence conjecture. As $i\to\infty$, the projections satisfy

$$\left\|\sin(2\pi m\mathord\cdot)-Q^q_{2i}\sin(2\pi m\mathord\cdot)\right\|\longrightarrow 0,$$

for $p=2i$, and

$$\left\|\cos(2\pi m\mathord\cdot)-Q^q_{2i+1}\cos(2\pi m\mathord\cdot)\right\|\longrightarrow 0,$$

for $p=2i+1$. The conjecture addresses the missing eigenfunction in the even-dimensional periodic spline case, where parity causes one of the two eigenfunctions at frequency $m$ to be orthogonal to the spline space.