Does an I²C IO Expander needs additional resistors for outputs? I have following setup (see also schematic in the end)

Arduino Nano is connected via I²C to the PCF8575C (link) IO Expander.
Both, SCL and SDA have 10kOhm pull-ups to 5V.
The Expander pins are connected to a MOSFET-driver that turns on/off 12V LED stripes.
The Expanders Bus address is set via Expanders A0-A2 pins to 0 (0x20).
The MOSFET driver pins have to be pulled to logic 1 (above 2.2V) in order to turn on the stripe.

Currently Arduino runs an blink example, that turns all Expander IO pins on/off every second (using Arduino Wire Library), but the MOSFET-driver does not turn on/off any Stripe.

I can manually turn on Stripes by connecting MOSFET-driver pins to Arduinos 5V (or 3.3V)

The power supply to the IO Expander is O.K.

By measuring the resistance between GND and IO pins of the Expander, I can see that it alternates every second (so I²C communication to the Expander works also)

Curious thing: While measuring this resistance (holding the multimeter test leads) to GND and to the IO pin of Expander, the MOSFET-driver turns the Stripes on and off! When I take the test leads away it stops working! Magic?!

What am I missing? Do the output pins of the PCF8575C need some extra pullup/pulldown resistors or something else?

Thank you in advance!

Schematics:
Yes, according to the datasheet (which should always be your first port of call for any chip) the outputs are Open Drain. That means you require a pullup resistor, exactly as if the outputs were simple buttons.