 Solution

Let the common on-site energy be zero and the common hopping magnitude be $t=|\beta |$. Consistent with the axial-$p$ convention and the formula in part (iii), the hopping signs are ${\beta }_s=-t$ and ${\beta }_p=+t$.

Without mixing,The curves cross at $k=\pi /(2a)$, $E=0$, with no gap. The lower-left panel shows the continuous pure-$s$ and pure-$p$ branches. Their identities remain fixed through the crossing; the energy-ordered lower state changes from $s$ to $p$ character on passing through it.