Conjecture on Theorem 2 for Artinian lex-ideals

Let $L$ be an Artinian lex-ideal in the polynomial ring $\mathbf{k}[x,y,z]$. The Boij–Söderberg decomposition of $L$ is the decomposition into pure diagrams described by Theorem 2. The conjecture. The statement of Theorem 2 holds for all Artinian lex-ideals in $\mathbf{k}[x,y,z]$. The claim extends the preceding result to the remaining cases and is stated without a proof in the supplied text.