Strong comparison principle for p-Laplacian type equations

Let $\Omega'$ be a bounded $C^{1,\alpha}$ subdomain compactly contained in a domain $\Omega\subset X$. Assume that the equation $Q'(w)=0$ admits a positive solution in $\Omega$, and let $u,v\in C^1(\Omega')\cap C(\overline{\Omega'})$ be nonnegative functions satisfying

$$Q'(v)\geq 0\quad\text{in }\Omega',\qquad Q'(u)\leq 0\quad\text{in }\Omega',\qquad u\leq v\quad\text{on }\partial\Omega'.$$

Strong comparison principle. Either $u<v$ throughout $\Omega'$, or $u=v$ throughout $\Omega'$. The weak comparison principle supplies the non-strict inequality $u\leq v$; the conjectured strict alternative is open for $p\ne2$, even for positive $p$-harmonic functions in $\mathbb{R}^d$ when $d>2$.