Determine the growth rate of the perturbed spin-1 level spacing

Prove or disprove that the magnitude of the one-sided derivatives $|\mu_1^{\pm}(L)|$ of the tracked $D_L$-th-to-$(D_L+1)$-st level spacing for the perturbed spin-1 Hamiltonian $H_1(\varepsilon,L)=\sum_x(P^{(0)}_{x,x+1}-\varepsilon P^{(1)}_{x,x+1})$ grows linearly with the chain length $L$ under open boundary conditions.

Background

The paper studies perturbations of the spin-1 singlet model on both sides of the purely biquadratic point and tracks the spacing between the $D_L$-th and $(D_L+1)$-st eigenvalues, where $D_L$ is the singlet-model ground-state degeneracy. Numerical data show that the magnitudes of the one-sided derivatives increase with system size.

For open boundaries, the authors explicitly conjecture linear growth in $L$. Establishing this asymptotic behavior would quantify the proposed instability of the singlet-model gap under arbitrarily small $SU(2)$-invariant perturbations.