I have not grown Giboshi, but have grown over 30 cultivars at some point or another, and below is my understanding, but I could well be wrong on some points.
I think Giboshi would hold some fruit for you without Pollination as “Parthenocarpic” fruit but probably set and hold more fruit and the fruit quality would likely be different, hopefully better if seeded.
The basics of Kaki persimmon pollination are:
Most Kaki are Parthenocarpic and will set at least some fruit without pollination, but might set more pollinated.
PCNA pollination constant non-astringent The fruit has minimal differences if seeded, possibly a little larger if seeded.
PCA pollination constant astringent The fruit will stay astringent whether seeded or not, the taste is astringent until the fruit softens up.
PVNA pollination variant non-astringent The fruit will be astringent unless seeded, and often the flesh surrounding the seed will color to a brownish and will be non-astringent even while firm. If not seeded the fruit will remain astringent until it softens. If not fully seeded you might end up with a partially astringent fruit.
PVA pollination variant astringent This is the one I am most unsure on. If the fruit remains astringent when pollinated/seeded how does it differ from PCA? I have always assumed there is a significant change in fruit quality hopefully for the better.
-Excessive nitrogen will cause fruit drop but I am not certain a normal lawn application would.
- Pollinated seeded fruit will hold on tree better, as well as changing the quality of the fruit on PV Pollination Variant cultivars.
-Other cultural influences such a drought and other stresses can cause fruit drop. I had a very dry may-July and dropped most my fruit this year on some cultivar.
-Tree age. Fruit drop on young trees is common if not the norm.
It’s unlikely your Native Virginiana would pollinate your Asian Kaki. There are hybrids of Kaki X Virginiana and some posts on them here, but the initial cross is not real easy as I understand. I did get seed set in my Nakita’s Gift hybrid with no effort not sure if male parent was a Kaki or Virginia from my woods, so at least some ensuing crosses at least pollinate easily.