Powers of 5 with no Zero in Decimal Representation

Unproven Hypotheses
The following powers of $5$ which contain no zero in their decimal representation are believed to be all that exist:
 * $1, 5, 25, 125, 625, 3125, 15 \, 625, 78 \, 125, 1 \, 953 \, 125, 9 \, 765 \, 625,$
 * $48 \, 828 \, 125, 762 \, 939 \, 453 \, 125, 3 \, 814 \, 697 \, 265 \, 625, 931 \, 322 \, 574 \, 615 \, 478 \, 515 \, 625,$
 * $116 \, 415 \, 321 \, 826 \, 934 \, 814 \, 453 \, 125, 34 \, 694 \, 469 \, 519 \, 536 \, 141 \, 888 \, 238 \, 489 \, 627 \, 838 \, 134 \, 765 \, 625$

but this has not been proven.

The corresponding indices are:
 * $0, 1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 17, 18, 30, 33, 58$