[seqfan] Re: Self-stuffable numbers

[Hans Havermann]

https://oeis.org/A322323

The appearance of 103008000000 at digit-size 12 was an unexpected delight. At digit-size 13 we have 1031008000000; at 14, 10311008000000; at 15, 103111008000000; and yes, the pattern continues!

Would understanding why this particular infinite family of self-stuffable numbers exists allow one to find (or rule out the existence of) other, larger examples?