Hmm. This is cool.... let me check if I understand it correctly. Also, this might help others understand in more laymens terms (if I got it right)

The Problem

You have coins sitting in a quantum-vulnerable address $A$, and you failed to move into a quantum-secure address before Q-day

The Solution

A soft fork that still allows people to spend from quantum-vulnerable addresses, but only with additional proof of ownership. The protocol relies on the existence of "knowledge asymmetries"... information that is hard for a quantum attacker to crack, but the true owner of the coins would have. We'll assume that such a knowledge asymmetry exists, and we'll call that secret $S$, which proves that you own $A$.

How it works

1. First, you create a new post-quantum keypair, $(Q,q)$. ($Q$ is the public key, $q$ is the private key).
2. You use $Q$ and $S$ to broadcast a "timestamp" transaction on block $T$. The purpose of this transaction is to prove that you knew $S$ at time $T$. $S$ and $Q$ can't be reversed from the transaction data.
3. After a delay of $d$ blocks, you publish a "rescue" transaction to move the funds from vulnerable address $A$ to post-quantum address $A^{\prime }$. The "rescue transaction" reveals $S$, thus proving that you control $A$, and it also contains proof that you published the previous timestamp transaction.

Reason for the timestamp and the delay

4. Once you send your rescue transaction to the mempool, $S$ is revealed. Without the timestamp/delay scheme, anyone can take that information and race you to moving the funds from $A$. However, because the protocol requires the proof of timestamp, the attacker would first have to create their own timestamp. But since you didn't reveal it until time $T+d$, they'd have to wait until at least $T+2d$ to steal your funds, because they'd first have to create a timestamp at time $T+d$ (when they see $S$), then wait another $d$ blocks.

Advantages of the approach

Allows people who failed to migrate to post-quantum addresses a chance to recover their funds, while at the same time not letting vulnerable coins become exposed to a quantum free-for-all.

Does that about catch it?

I think your article has a lot more details about miner incentives and other implementation details, but I just tried to capture the high level gist of the idea.