From On the $q$-Strong Diffie-Hellman Problem, the following problem is well-known to be hard.
For an randomly chosen element $x \in \mathbb{Z}_p$ and a random generator $g \in \mathbb{G}$, the exponent $q$-strong Diffie-Hellman Problem is, given $(g, g^x, g^{x^2}, \dots, g^{x^{q-1}}) \in \mathbb{G}^{q}$ to compute an element $g^{x^q} \in \mathbb{G}$.
Here, I want to ask how hard is the following problem:
For an randomly chosen element $x \in \mathbb{Z}_p$ and a random generator $g \in \mathbb{G}$, given $(g, g^x, g^{x^2}, \dots, g^{x^{q-1}}, g^{x^{q+1}}, g^{x^{q+2}}, \dots, g^{x^{2q-2}}) \in \mathbb{G}^{2q-2}$ to compute an element $g^{x^q} \in \mathbb{G}$.
Can it be reduced to other well-known Diffie-Hellman problem?