Asymptotic polynomial conjecture for numerical semigroups generated by consecutive powers

Let $R_n^k$, with $n=T_km+j$, be a numerical semigroup given by its minimal relations on the residue class of $n$ modulo $T_k$:

$$R_{T_km+j}^k: \left(\begin{array}{rrr}E_{11}^{(k)}(m)&-E_{12}^{(k)}(m)&-E_{13}^{(k)}(m)\\-E_{21}^{(k)}(m)&E_{22}^{(k)}(m)&-E_{23}^{(k)}(m)\\-E_{31}^{(k)}(m)&-E_{32}^{(k)}(m)&E_{33}^{(k)}(m)\end{array}\right), \qquad j\leq \frac{T_k}{2}.$$

If $k=2q$, the polynomials $E_{ij}^{(k)}(m)$ have degree $q$:

$$E_{ij}^{(2q)}(m)=A_{ij}m^q+B_{ij}m^{q-1}+\ldots+C_{ij}m+D_{ij}, \qquad 1\leq i,j\leq 3,$$

and if $k=2q+1$, then $E_{ij}^{(2q+1)}(m)$ has degree $q+1$ for $(i,j)=(1,1),(1,2),(2,1),(2,2)$ and degree $q$ for $(i,j)=(1,3),(2,3),(3,1),(3,2),(3,3)$. Asymptotic polynomial conjecture. For even $k=2q$, the Frobenius number and genus satisfy $F(n),G(n)={\cal O}(n^{3q})$. For odd $k=2q+1$, they satisfy $F(n),G(n)={\cal O}(n^{3q+2})$. The conjecture proposes a uniform polynomial description of the minimal relations and corresponding growth bounds for Frobenius numbers and genera of these numerical semigroups. The supplied text does not indicate whether the claim has been proved or remains open.