No uniform realisation is possible.

Incidence matrix according to Dynkin symbol

β3β3β3x

both( . . . . ) | 120 |  1  1   2   2  1 |  1  1  1   3  2  2  2 |  1  1  1 1  3
----------------+-----+------------------+-----------------------+--------------
both( . . . x ) |   2 | 60  *   *   *  * |  0  0  1   0  2  1  1 |  0  1  1 1  2
both( s 2 s . ) |   2 |  * 60   *   *  * |  0  0  0   2  0  2  0 |  1  0  1 0  2
sefa( s3s . . ) |   2 |  *  * 120   *  * |  1  0  0   1  1  0  0 |  1  1  0 0  1
sefa( . s3s . ) |   2 |  *  *   * 120  * |  0  1  0   1  0  0  1 |  1  0  0 1  1
sefa( . . β3x ) |   2 |  *  *   *   * 60 |  0  0  1   0  0  1  1 |  0  0  1 1  1
----------------+-----+------------------+-----------------------+--------------
both( s3s . . ) ♦   3 |  0  0   3   0  0 | 40  *  *   *  *  *  * |  1  1  0 0  0
both( . s3s . ) ♦   3 |  0  0   0   3  0 |  * 40  *   *  *  *  * |  1  0  0 1  0
      . . β3x   ♦   6 |  3  0   0   0  3 |  *  * 20   *  *  *  * |  0  0  1 1  0
sefa( s3s3s . ) |   3 |  0  1   1   1  0 |  *  *  * 120  *  *  * |  1  0  0 0  1
sefa( s3s . x ) |   4 |  2  0   2   0  0 |  *  *  *   * 60  *  * |  0  1  0 0  1
sefa( β 2 β3x ) |   4 |  1  2   0   0  1 |  *  *  *   *  * 60  * |  0  0  1 0  1
sefa( . β3β3x ) |   4 |  1  0   0   2  1 |  *  *  *   *  *  * 60 |  0  0  0 1  1
----------------+-----+------------------+-----------------------+--------------
both( s3s3s . ) ♦  12 |  0  6  12  12  0 |  4  4  0  12  0  0  0 | 10  *  * *  *
both( s3s 2 x ) ♦   6 |  3  0   6   0  0 |  2  0  0   0  3  0  0 |  * 20  * *  *
      β 2 β3x   ♦  12 |  6  6   0   0  6 |  0  0  2   0  0  6  0 |  *  * 10 *  *
      . β3β3x   ♦  24 | 12  0   0  24 12 |  0  8  4   0  0  0 12 |  *  *  * 5  *
sefa( β3β3β3x ) ♦   6 |  2  2   2   2  1 |  0  0  0   2  1  1  1 |  *  *  * * 60

starting figure: x3x3x3x