During Apollo 13, an explosion in the service module caused the command module to lose electrical power. This required using the environmental control system of the lunar module to remove carbon dioxide from the air, instead of the one in the command module. However, CO$_2$ levels continued to rise.
The movie Apollo 13 explains the problem as such:
FRED HAISE: (expletive) I know why my numbers are wrong. I only figured it for two people.
JACK SWIGERT: Maybe I should just hold my breath.
NEWS ANCHOR (on TV): Well, the astronauts appear to have enough oxygen to keep them alive. One thing they have too much of is carbon dioxide. With each breath that three men exhale more of the poisonous gas into the lunar module cockpit and the scrubbers intended to keep the atmosphere breathable are quickly becoming saturated.
The movie suggests that the CO$_2$ problem was a limited rate. In brief, they "can't keep up." That is, the LiOH canisters can only filter so many moles of CO$_2$ per minute. A normal LM crew of 2 produces CO$_2$ below this rate, so the filter keeps up. Apollo 13's crew of 3 produced CO$_2$ above this rate, the canisters "saturated", and the excess CO$_2$ accumulated in the cabin.
However, another possibility is that the LiOH canisters have a limited capacity. In brief, they "run out." They can only filter so many total moles of CO$_2$. More astronauts means more total moles of CO$_2$ to be handled by the LM. Once a canister's limit is reached, no more CO$_2$ can be filtered, and the CO$_2$ level rises until a new canister is installed. The CO$_2$ level could even deliberately be allowed to rise in order to ration the canister supply.
Which of these factors was more important in causing the CO$_2$ levels to rise during Apollo 13? (It's possible that both play a role, but I am looking for which one was more relevant to the Apollo 13 situation.)
Please support your answer with official NASA sources or published scientific research. Answers based solely on speculation, the movie, or other popular accounts are not acceptable.