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How does the transition width of a FIR filter relate to phase delay at the output of the filter?

I have been trying to find information on the subject for days.

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In this answer I gave two well-known heuristic design formulas relating the filter order of (linear-phase) FIR low-pass filters to the transition bandwidth. According to both formulas, the transition bandwidth is inversely proportional to the filter order. Since the phase delay of a linear-phase FIR filter equals $M/2$, where $M$ is the filter order, the phase delay $\tau_p$ and the transition bandwidth $\Delta\omega$ are also inversely proportional to each other:

$$\Delta\omega\propto \frac{1}{\tau_p}$$

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