Suppose a green card holder arrived on a flight to the US, but at some point during the flight, lost their green card and therefore could not present it to CBP. It seems that this person would be an inadmissible alien since they fail to meet the documentary requirements under INA 212(a)(7).
There are some ways out of this situation: the inspecting officer could waive the documentary requirement (INA 211(b)), grant deferred inspection (8 CFR 235.2), or permit the alien to withdraw their application for admission (INA 235(a)(4)). However, all of those require an exercise of discretion.
Therefore, it appears that if DHS is really determined to deport the arriving alien in this situation, they can refuse to grant a waiver, refuse to grant deferred inspection, refuse to grant withdrawal, and place the alien in removal proceedings. In removal proceedings, the immigration judge could similarly deny all forms of relief and issue an order of removal. The alien would then lose their LPR status and be banned from the US for 5 years.
Legally speaking, are there a way for such an arriving alien (a returning LPR, who has previously complied with all immigration laws, has not abandoned their residence, and has merely lost their green card on the way to the port of entry) to avoid deportation that does not require a favourable exercise of discretion?