= Solution
In the orthogonal nearest-neighbour <tight-binding model>,
$$
\alpha_s=\langle s_j|H|s_j\rangle,\qquad
\beta_s=\langle s_j|H|s_{j+1}\rangle.
$$
The on-site integral $\alpha_s$ sets the band centre, while the <hopping integral> $\beta_s$ describes coupling between neighbouring orbitals. With a consistent phase convention, the bonding combination at the zone centre has $\beta_s<0$.
For <p orbitals> directed along the chain, use
$$
\boxed{E_p(k)=\alpha_p+2\beta_p\cos(ka),\qquad \beta_p>0.}
$$
Here every positive $p$ lobe points in the same spatial direction. Neighbouring equal-sign coefficients then bring opposite-sign facing lobes together and make the zone-centre combination antibonding; the alternating zone-boundary combination is bonding. Thus the usual axial $p$ hopping has the opposite sign to $s$ hopping. <Orbital phase and the sign of one-dimensional hopping> must be treated consistently; replacing only $\alpha_s$ by $\alpha_p$ while silently keeping the same physical hopping sign would give the wrong band ordering.
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