Dots and Cubes

Suppose your friend Glenn creates a 3D version of dots and boxes called "Dots and Cubes," and wants you to play with him. The new rules are as follows:

• Each player takes turns connecting one dot to another (no diagonals, only the 3D grid).
• Players may not play on existing connections.
• Players can only claim a cube if they're the last to complete the $12$ connections required to form a cube.
• If a player claims a cube on their turn, they get to go again.
• The game ends once all legal connections are made.
• The player with the most claims wins.

Below is a $2\times 1 \times 1$ grid of Dots and Cubes with the $\color{#D61F06}\text{red}$ player claiming the front cube:

You and Glenn both play defensively (meaning, if possible, you won't play on an $n$ -dimensional square with 2 less connections needed to claim it). Glenn notices that the second player always wins.

Glenn says any $n$ -dimensional Dots and Boxes $($ where $n \geq{2}$ and the dots and boxes can be represented by an $n$ -dimensional grid of $a_1\times a_2\times \cdots \times a_n)$ will always have the second player win if both players are playing defensively .

Is this true?