Conjectural $L^2$ convergence rate for multidimensional convection-diffusion collocation

Let $R$ and $R_0$ be the domains defined in Section 2. Let $u\in C^{m+2,\alpha}(R_0)$, $k\in [C^{m,\alpha}(R_0)]^d$, $g\in C^{m+2,\alpha}(\overline{R_0})$, and $f\in C^{m,\alpha}(R_0)$, with $\alpha\in(0,1)$, and suppose that they satisfy

$$-\Delta u+k\cdot\nabla u=f\quad\text{in }R,\qquad u=g\quad\text{on }\partial R.$$

Multidimensional convergence conjecture. There exists a constant $C=C(R,k,m,|u|_{H^{m+2}(R_0)})$ such that

$$|u-u_N|_{L^2}\leq CN^{-m+\sigma}$$

for some fixed $\sigma>0$. This is presented as a consequence of the preceding Lebesgue-constant conjecture; the source gives no proof or resolution of the asserted rate.