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Claim. Every graph $G$ with maximum degree $\Delta(G)\le3$, multiple edges allowed, has a strong edge-coloring with at most $10$ colors, so $\mathrm{sq}(G)\le10$. This is Theorem 1 (p. 250) of L. D. Andersen, The strong chromatic index of a cubic graph is at most 10, Discrete Math. 108 (1992), no. 1--3, 231--252, paged at theorem_1 of the source card; the theorem also gives a linear time algorithm. Since $10\le\lfloor\frac54\cdot9\rfloor=11$, the bound of Problem 149 holds for every graph with $\Delta=3$. For $\Delta\le2$ the bound $10$ does not give it (the conjectured bound is $5$ when $\Delta=2$), but those graphs are paths and cycles, for which it is elementary (the 1993 paper's p. 152), so the statement holds for every $\Delta\le3$. The paper's graph $G_{10}$ (p. 232), a five-cycle with two consecutive vertices doubled, shows that $10$ is attained. Horák, He and Trotter proved the same bound independently, on their claim page; Andersen's p. 231 records their proof as a private communication.

Covers. Every graph of maximum degree at most $3$ (the instances $\Delta\le3$ of the statement; for $\Delta\le2$ the graphs are paths and cycles and the bound is elementary). The theorem says nothing about $\Delta\ge4$, where the question is open.

Depends on. Nothing in this wiki; the proof is self-contained apart from Hall's theorem.

Acceptance. Refereed: Discrete Mathematics (the Crossref record gives volume 108, issue 1--3, pp. 231--252, issue dated October 1992; the day is the issue's nominal first day, used for this page's date). The site's curator credits the result in the problem's commentary, but the site labels the problem OPEN, so the commentary is not listed as review. Read depth: Theorem 1, the definitions and the $G_{10}$ passage are checked clause by clause, and the proof of Theorem 1 (pp. 250--251) with its reduction to Lemmas 2--15; the lemmas' case analyses are not checked, so nothing is independently reviewed in this repository.