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Try $(x=28, y=42, z=84, w=98)$ and $(x=13, y=84, z=204, w=221)$.

EDIT: It was a coincidence. The following are some more solutions:

$$\matrix{ \left\{ 15,78,3570,6246 \right\} ,& \left\{ 21,60,453,900 \right\} ,& \left\{ 21,60,900,1797 \right\} ,\cr \left\{ 29,35,11670, 13128 \right\} ,& \left\{ 30,42,69,90 \right\} ,& \left\{ 30,42,90,156 \right\} ,\cr \left\{ 30,42,156,300 \right\} ,& \left\{ 30,42,300,594 \right\} , &\left\{ 30,42,594,1185 \right\}\cr} $$

Try $(x=28, y=42, z=84, w=98)$ and $(x=13, y=84, z=204, w=221)$.

EDIT: It was a coincidence. The following are some more solutions that have no $84$:

$$\matrix{ \left\{ 15,78,3570,6246 \right\} ,& \left\{ 21,60,453,900 \right\} ,& \left\{ 21,60,900,1797 \right\} ,\cr \left\{ 29,35,11670, 13128 \right\} ,& \left\{ 30,42,69,90 \right\} ,& \left\{ 30,42,90,156 \right\} ,\cr \left\{ 30,42,156,300 \right\} ,& \left\{ 30,42,300,594 \right\} , &\left\{ 30,42,594,1185 \right\}\cr} $$

Try $(x=28, y=42, z=84, w=98)$ and $(x=13, y=84, z=204, w=221)$.

EDIT: It was a coincidence. The following are the solutions for four distinct positive integers with at least two $< 84$:

$$\matrix{ \left\{ 15,78,3570,6246 \right\} ,& \left\{ 21,60,453,900 \right\} ,& \left\{ 21,60,900,1797 \right\} ,\cr \left\{ 29,35,11670, 13128 \right\} ,& \left\{ 30,42,69,90 \right\} ,& \left\{ 30,42,90,156 \right\} ,\cr \left\{ 30,42,156,300 \right\} ,& \left\{ 30,42,300,594 \right\} , &\left\{ 30,42,594,1185 \right\}\cr} $$

Try $(x=28, y=42, z=84, w=98)$ and $(x=13, y=84, z=204, w=221)$.

EDIT: It was a coincidence. The following are some more solutions:

$$\matrix{ \left\{ 15,78,3570,6246 \right\} ,& \left\{ 21,60,453,900 \right\} ,& \left\{ 21,60,900,1797 \right\} ,\cr \left\{ 29,35,11670, 13128 \right\} ,& \left\{ 30,42,69,90 \right\} ,& \left\{ 30,42,90,156 \right\} ,\cr \left\{ 30,42,156,300 \right\} ,& \left\{ 30,42,300,594 \right\} , &\left\{ 30,42,594,1185 \right\}\cr} $$

Try $(x=28, y=42, z=84, w=98)$ and $(x=13, y=84, z=204, w=221)$.

Try $(x=28, y=42, z=84, w=98)$ and $(x=13, y=84, z=204, w=221)$.

EDIT: It was a coincidence. The following are the solutions for four distinct positive integers with at least two $< 84$:

$$\matrix{ \left\{ 15,78,3570,6246 \right\} ,& \left\{ 21,60,453,900 \right\} ,& \left\{ 21,60,900,1797 \right\} ,\cr \left\{ 29,35,11670, 13128 \right\} ,& \left\{ 30,42,69,90 \right\} ,& \left\{ 30,42,90,156 \right\} ,\cr \left\{ 30,42,156,300 \right\} ,& \left\{ 30,42,300,594 \right\} , &\left\{ 30,42,594,1185 \right\}\cr} $$