No uniform realisation is possible.

Incidence matrix according to Dynkin symbol

x3β3β3x

both( . . . . ) | 120 |  1  1  1   2  1 |  1  1  1  1  2  1  1  2 | 1  1  1 1  2
----------------+-----+-----------------+-------------------------+-------------
both( x . . . ) |   2 | 60  *  *   *  * |  1  1  0  0  1  0  1  0 | 1  1  1 0  1
both( . . . x ) |   2 |  * 60  *   *  * |  1  0  0  1  0  1  0  1 | 0  1  1 1  1
both( x3β . . ) |   2 |  *  * 60   *  * |  0  1  0  0  1  1  0  0 | 1  1  0 0  1
sefa( . s3s . ) |   2 |  *  *  * 120  * |  0  0  1  0  1  0  0  1 | 1  0  0 1  1
sefa( . . β3x ) |   2 |  *  *  *   * 60 |  0  0  0  1  0  0  1  1 | 0  0  1 1  1
----------------+-----+-----------------+-------------------------+-------------
both( x . . x ) |   4 |  2  2  0   0  0 | 30  *  *  *  *  *  *  * | 0  1  1 0  0
      x3β . .   ♦   6 |  3  0  3   0  0 |  * 20  *  *  *  *  *  * | 1  1  0 0  0
both( . s3s . ) ♦   3 |  0  0  0   3  0 |  *  * 40  *  *  *  *  * | 1  0  0 1  0
      . . β3x   ♦   6 |  0  3  0   0  3 |  *  *  * 20  *  *  *  * | 0  0  1 1  0
sefa( x3β3β . ) |   4 |  1  0  1   2  0 |  *  *  *  * 60  *  *  * | 1  0  0 0  1
sefa( x3β . x ) |   4 |  0  2  2   0  0 |  *  *  *  *  * 30  *  * | 0  1  0 0  1
sefa( x . β3x ) |   4 |  2  0  0   0  2 |  *  *  *  *  *  * 60  * | 0  0  1 0  1
sefa( . β3β3x ) |   4 |  0  1  0   2  1 |  *  *  *  *  *  *  * 60 | 0  0  0 1  1
----------------+-----+-----------------+-------------------------+-------------
      x3β3β .   ♦  24 | 12  0 12  24  0 |  0  4  8  0 12  0  0  0 | 5  *  * *  *
      x3β . x   ♦  12 |  6  6  6   0  0 |  3  2  0  0  0  3  0  0 | * 10  * *  *
      x . β3x   ♦  12 |  6  6  0   0  6 |  3  0  0  2  0  0  3  0 | *  * 10 *  *
      . β3β3x   ♦  24 |  0 12  0  24  0 |  0  0  8  4  0  0  0 12 | *  *  * 5  *
sefa( x3β3β3x ) ♦   8 |  2  2  2   4  2 |  0  0  0  0  2  1  1  2 | *  *  * * 30

or
both( . . . . )   | 120 |   2   2   2 |  1  2  1   4  2 |  2  2  2
------------------+-----+-------------+-----------------+---------
both( x . . . ) & |   2 | 120   *   * |  1  1  0   1  1 |  1  2  1
sefa( x3β . . ) & |   2 |   * 120   * |  0  1  0   1  1 |  1  1  1
sefa( . s3s . )   |   2 |   *   * 120 |  0  0  1   2  0 |  2  0  1
------------------+-----+-------------+-----------------+---------
both( x . . x )   |   4 |   4   0   0 | 30  *  *   *  * |  0  2  0
      x3β . .   & ♦   6 |   3   3   0 |  * 40  *   *  * |  1  1  0
both( . s3s . )   ♦   3 |   0   0   3 |  *  * 40   *  * |  2  0  0
sefa( x3β3β . ) & |   4 |   1   1   2 |  *  *  * 120  * |  1  0  1
sefa( x3β . x ) & |   4 |   2   2   0 |  *  *  *   * 60 |  0  1  1
------------------+-----+-------------+-----------------+---------
      x3β3β .   & ♦  24 |  12  12  24 |  0  4  8  12  0 | 10  *  *
      x3β . x   & ♦  12 |  12   6   0 |  3  2  0   0  3 |  * 20  *
sefa( x3β3β3x )   ♦   8 |   4   4   4 |  0  0  0   4  2 |  *  * 30

starting figure: x3x3x3x