Yes, the depth of the warm layer increases rather quickly (see the cross-section animation below). The surface is usually the slowest to warm in these types of events with strong HP/ageostrophic low-level flow of cold air.
Guidance from the 06z 3-km NAM shows forcing along the frontal surface decreasing in height with time; the greatest frontogenesis over our area is around 600 mb at ~12z, and the frontogenesis weakens and decreases in height with time (tracing the frontal surface). The dry air present at the low levels ahead of the precipitation makes me question if the more intense (>1-2"/hr) snowfall rates will occur beyond a brief period, especially as the forcing weakens after 12z.
By the way, the cross-section feature from Tropical Tidbits is fantastic!