Equalise panel height in mutliplot figure using ggplot2 and grid.arrange - r

Question

How can I equalise the height of the grey panel and at the same time align all panels (even the flipped and reversed)? The problem is that due to the fact that the top plot does have no axis labels etc, the panel height is slightly larger.
library(ggplot2)
library(gridExtra)
library(grid)
# Plot 1
df1 <- data.frame(n = 1:10, y = rnorm(10, 5))
plot1 <- ggplot(df1, aes(x = n, y = y)) +
geom_bar(stat = "identity") +
labs(y = 'AU', x = NULL) +
theme(axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.ticks.x = element_blank(),
plot.margin = unit(c(0.1, 0.1, 0.1, 0), "cm"))
# Plot 2
df2 <- data.frame(x = rnorm(100), y = rnorm(100))
plot2 <- ggplot(df2, aes(x = x, y = y)) +
geom_point() +
labs(y = 'AU', x = NULL) +
theme(axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.ticks.x = element_blank(),
plot.margin = unit(c(0.1, 0.1, 0.1, 0), "cm"))
# Plot 3
df3 <- data.frame(x = rnorm(100))
plot3 <- ggplot(df3, aes(x = x)) +
coord_flip() +
geom_histogram() +
theme(axis.title.y = element_blank(),
axis.text.y = element_blank(),
axis.ticks.y = element_blank(),
axis.title.x = element_blank(),
axis.text.x = element_blank(),
axis.ticks.x = element_blank(),
plot.margin = unit(c(0.1, 0.1, 0.1, 0), "cm"))
# Plot 4
df4 <- data.frame(n = 1:10, y = rnorm(10, 4))
plot4 <- ggplot(df4, aes(x = n, y = y)) +
scale_y_reverse() +
geom_bar(stat = "identity") +
labs(y = 'AU', x = 'N') +
theme(plot.margin = unit(c(0.1, 0.1, 0.1, 0), "cm"))
# Get blank plot
blank <- grid.rect(gp=gpar(col="white"))
# Combine everything into one plot
## Get Grobs
gplot1 <- ggplotGrob(plot1)
gplot2 <- ggplotGrob(plot2)
gplot3 <- ggplotGrob(plot3)
gplot4 <- ggplotGrob(plot4)
# Align widths
maxWidth = grid::unit.pmax(gplot1$widths, gplot2$widths, gplot4$widths)
gplot1$widths <- as.list(maxWidth)
gplot2$widths <- as.list(maxWidth)
gplot4$widths <- as.list(maxWidth)
# Grid arrange
grid.arrange(grobs = list(gplot1, blank, gplot2, gplot3, gplot4, blank),
newpage = FALSE,
ncol = 2,
widths = c(12, 3))
Since I want the middle scatter plot's y-axis to be aligned to the flipped x-axis of the righthand side plot, I can't see how I can use rbind() as suggested here would work for here.
Also, setting the grob heights manually, e.g.
gplot4$heights<- gplot1$heights
just cuts of the labels. While, the reverse order (gplot1$heights<- gplot4$heights) introduces a big gap between the top and the middle plot.