Conjecture on perfect powers in alternating sums of consecutive cubes

Let $r=2d$ with $1\leq d\leq 50$, let $p\geq 5$ be prime, and consider the equation

$$(x+1)^3-(x+2)^3+\cdots-(x+r)^3+(x+r+1)^3=z^p.$$

Here $d,x,z$ are integers. Alternating-cube perfect-power conjecture. The possible integer solutions are

$$(d,x,z,p)\in\{(2,2,-2,7),(3,12,-3,7),(6,14,-6,5),(27,215,-9,7)\}.$$

This conjecture concerns perfect powers represented by alternating sums of an even number of consecutive cubes. The supplied text gives examples and restricts $d\leq 50$, but gives no resolution status for the asserted list.