Subquartic saturation-number conjecture

Let $G$ be a connected simple graph, and let $\mu^{*}(G)$ denote its saturation number, the minimum cardinality of a maximal matching. Let

$$H(G)=\sum_{\{u,v\}\in E(G)}\frac{2}{d(u)+d(v)}$$

be its harmonic index. A graph is subquartic when its maximum degree is at most four. Subquartic saturation-number conjecture. Every connected subquartic graph satisfies

$$\mu^{*}(G)\leq H(G).$$

The inequality has been verified by exhaustive search for subquartic graphs with at most eleven vertices, but the source presents the assertion as a conjecture for the whole class. Its status beyond the checked finite range remains open.