Restricted smallest-root conjecture for incomplete positive linear recurrence sequences

Let $L\in\mathbb{Z}_{>0}$, let

$$N_L=\left\lceil\frac{L(L+1)}{4}\right\rceil,$$

and let $\lambda_L$ be the principal root of

$$x^L-x^{L-1}-N_L-1.$$

Restricted smallest-root conjecture. If the sequence generated by $[c_1,\ldots,c_L]$ is incomplete and satisfies

$$\sum_{i=1}^{L}c_i\leq \left\lceil\frac{L(L+1)}{4}\right\rceil+2,$$

then its principal root is at least $\lambda_L$. This is a restricted version of the proposed lower bound for incomplete sequences; the source provides motivation and partial analysis but no proof of the stated claim.