[Kawaii Ryûkishi]

I figured it out: if you save or enter the travel agency, the counter goes back to zero. I'll go back for the sigil after I acquire the ability to disable random encounters, which interrupt the whole process greatly.

Incidentally, here are the prizes I did receive:
5 times.......X-Potion(2)
10 times.....Mega-Potion(2)
20 times.....MP Sphere(2)
50 times.....Strength Sphere(3)
100 times...HP Sphere(3)
150 times...Megalixir(4)

Thanks, everybody.