Q: iPhone says headphones connected when they aren't

I have a Verizon iPhone 6 with up-to-date iOS 9.3.2 software.

Today I found that, when answering a phone call, the caller could hear me speaking through the iPhone's microphone, but I could not hear any sound through the iPhone's speaker when holding it up to my head. However, if I switched it to speakerphone, I could hear the caller's voice normally.

When I press the volume up/down buttons, the window that appears showing the volume bar is labeled "Headphones" at the top.

If I plug in headphones, I can hear phone calls and audio from other apps normally. However, if I unplug the headphones, it still seems to think that they are connected.

I don't think Bluetooth is an issue, because I've tried turning off Bluetooth and turning off any nearby Bluetooth devices.

Thinking it might be a physical problem with the headphone jack, I've plugged and unplugged headphones several times, but this doesn't help.

Per recommendations in a help file here, I've tried restarting the phone, but it doesn't help. As noted above, the software is up-to-date.

Does this sound like a hardware failure with the headphone jack? Or is there anything else I can try?