Conjecture: there is only one configuration of $d$ lines in $n$ dimensional space where any pair have same angle and $d$ is the max possible.

Question

The sequence here: http://oeis.org/A002853 tells us the maximum number of lines that can be placed in $d$ dimensional space in a way that any pair of lines has the same angle as any other pair and all of them pass through the origin. In general, there can be multiple configurations with $n$ lines that satisfy this property. For example, when $d=3$ and we want to place $3$ lines, we can have the coordinate system ($x$, $y$ and $z$ axes) or three lines at $3 \pi/2$ radians restricted to a plane. So, there are at least two such configurations.

I conjecture that when we have the maximum possible number of lines, there will be only one configuration possible. Is it possible to prove this conjecture or refute with counter-example?

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Related:

1. Number of configurations of lines in $d$ dimensional space where any pair of them have the same angle.
2. How to draw six lines in 3-d space such that angle between any pair of them is the same.