You can insert a copy of the axes inside the gtable,
library(gtable)
g <- ggplotGrob(p)
# locate the panels
panels <- grep("panel", g$layout$name)
top <- unique(g$layout$t[panels])
# intersperse a copy of the bottom axes
all <- gtable:::rbind_gtable(gtable:::rbind_gtable(g[seq.int(min(top)), ],
g[max(top)+1,], "first"),
g[seq(min(top)+1, nrow(g)),], "first")
grid.newpage()
grid.draw(all)