The generalized lock inequality for positive words

Let $w$ be a positive word of the form

$$w=a^{\alpha_1}b^{\beta_1}\cdots a^{\alpha_m}b^{\beta_m}.$$

Suppose that $R(w,r,s)=p/q$, where $p/q$ is reduced. Generalized lock inequality. Then

$$\left|\frac{p}{q}-h_a(w)r-h_b(w)s\right|\leq \frac{m}{q}.$$

For the linear representation with $a=R_r$ and $b=R_s$, one has $\operatorname{rot}^\sim(w)=h_a(w)r+h_b(w)s$; thus the conjecture asserts that extremal representations become closer to the linear representation as the denominator of $R(w,r,s)$ increases. It generalizes the lock inequality and is supported by experimental evidence.