The conjecture on the validity of relation (6.24') for one-dimensional systems

Let $C$ and $u$ be the parameters appearing in relation (6.24'), and let $L_s$ denote the function used there. For $n=1$ and $u<0$, assume

$$-C\nu\leq 1.$$

Relation (6.24') is the inequality established in the preceding discussion for the corresponding Bose and Fermi cases. Conjecture on relation (6.24'). Relation (6.24') should hold for all $-C\nu\leq 1$ when $n=1$ and $u<0$. The \text further verifies the extremal case $C\nu=-1$ by estimating $L_{3/2}(-1)/L_{1/2}(-1)>1$, but does not establish the full asserted range.