Wallet pop-up does not show if both MetaMask and Phantom are installed

Steps to reproduce:

* Install and enable both MetaMask and Phantom. (This was attempted in MS Edge, but should be the same for all Chromium-based browsers)
* Open this URL: https://chainsafe.github.io/game-web3wallet/?action=sign&message=hello

Expected behaviour:

The user should be asked to select a wallet, or in the worst case one should be shown be default.

Actual behaviour:

The wallets do not show at all.

Workarounds:

Disabling one of the wallets causes the other one to show up.