Using .SD with update by reference

Reproduce the supplied Products and ProductPriceHistory example in R, then inspect the [.data.table update-by-reference and .SDcols path implicated by the error. Determine whether the requested .SD form requires a behavior change or documentation, add coverage for the chosen outcome, and verify the existing join example remains unchanged.

First, let me introduce the current example in https://rdatatable.gitlab.io/data.table/articles/datatable-joins.html#updating-by-reference

library(data.table)
Products = data.table(
  id = c(1:4,
         NA_integer_),
  name = c("banana",
           "carrots",
           "popcorn",
           "soda",
           "toothpaste"),
  price = c(0.63,
            0.89,
            2.99,
            1.49,
            2.99),
  unit = c("unit",
           "lb",
           "unit",
           "ounce",
           "unit"),
  type = c(rep("natural", 2L),
           rep("processed", 3L))
)

Products
#       id       name price   unit      type
#    <int>     <char> <num> <char>    <char>
# 1:     1     banana  0.63   unit   natural
# 2:     2    carrots  0.89     lb   natural
# 3:     3    popcorn  2.99   unit processed
# 4:     4       soda  1.49  ounce processed
# 5:    NA toothpaste  2.99   unit processed

ProductPriceHistory = data.table(
  product_id = rep(1:2, each = 3),
  date = rep(as.IDate(c("2024-01-01", "2024-02-01", "2024-03-01")), 2),
  price = c(0.59, 0.63, 0.65,  # Banana prices
            0.79, 0.89, 0.99)  # Carrot prices
)

ProductPriceHistory
#    product_id       date price
#         <int>     <IDat> <num>
# 1:          1 2024-01-01  0.59
# 2:          1 2024-02-01  0.63
# 3:          1 2024-03-01  0.65
# 4:          2 2024-01-01  0.79
# 5:          2 2024-02-01  0.89
# 6:          2 2024-03-01  0.99

copy(Products)[ProductPriceHistory,
               on = .(id = product_id),
               j = `:=`(price = tail(i.price, 1),
                        last_updated = tail(i.date, 1)),
               by = .EACHI][]
#       id       name price   unit      type last_updated
#    <int>     <char> <num> <char>    <char>       <IDat>
# 1:     1     banana  0.65   unit   natural   2024-03-01
# 2:     2    carrots  0.99     lb   natural   2024-03-01
# 3:     3    popcorn  2.99   unit processed         <NA>
# 4:     4       soda  1.49  ounce processed         <NA>
# 5:    NA toothpaste  2.99   unit processed         <NA>

I want to do this, but with price and date inside .SD, but I'm getting errors:

Products[ProductPriceHistory, on = .(id = product_id), c("price", "last_updated") := lapply(.SD, tail, 1), .SDcols = c("i.price", "i.date"), by = .EACHI]
Error in `[.data.table`(Products, ProductPriceHistory, on = .(id = product_id),  : 
  Some items of .SDcols are not column names: [i.price, i.date]

I tried other possibilities (removing i. of column names in .SDcols, using i.SD, i..SD...), with similar results.

How may I proceed?

Thanks!