Well, I don't know if I'm supposed to, but since I found a solution, I'll write the general idea here.

[This is from an unpublished manuscript by P. Smith, the first author of the paper]: If $A$ is CM, let $\omega_A = H^d_\mathfrak m(A)^*$ be its dualizing module. Then there is a spectral sequence $$ E^{pq}_2 = \underline{Ext}^p_A(\underline{Ext}^q(M, \omega_A),\omega_A) \Rightarrow \begin{cases} M \mbox{ if p = q} \\ 0 \mbox{ otherwise}\end{cases}$$

Since $H_\mathfrak{m}^i(M)^* \cong \underline{Ext}_A^i(M,\omega_A)$, the convergence of this SS to a non-zero result guarantees that there must be a non-zero local cohomology module (and in fact, that there is a non-zero one with $i \leq d$)

Well, I don't know if I'm supposed to, but since I found a solution, I'll write the general idea here.

[This is from an unpublished manuscript by P. Smith, the first author of the paper]: If $A$ is CM, let $\omega_A = H^d_\mathfrak m(A)^*$ be its dualizing module. Then there is a spectral sequence $$ E^{pq}_2 = \underline{Ext}^p_A(\underline{Ext}^q(M, \omega_A),\omega_A) \Rightarrow \begin{cases}M&\mbox{ if p = q} \\\ 0 &\mbox{ otherwise}\end{cases}$$

Since $H_\mathfrak{m}^i(M)^* \cong \underline{Ext}_A^i(M,\omega_A)$, the convergence of this SS to a non-zero result guarantees that there must be a non-zero local cohomology module (and in fact, that there is a non-zero one with $i \leq d$)